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Improved Rank Aggregation under Fairness Constraint

Diptarka Chakraborty, Himika Das, Sanjana Dey, Alvin Hong Yao Yan

TL;DR

This work advances fair rank aggregation under proportional fairness by delivering a $(2+\varepsilon)$-approximation through a two-stage approach that reduces the problem to a colorful bi-partition on a weighted tournament, then applies PTAS-based ranking on each partition. It also introduces a generic $2.881$-approximation algorithm that works with any fairness notion provided a closest fair ranking can be computed, enabling robust guarantees across stricter fairness notions like block fairness. Empirical evaluation on football and Movielens datasets shows substantial improvements over prior baselines and confirms the practical effectiveness of the proposed methods. Overall, the paper provides theoretically tight ideas and scalable algorithms that significantly improve fair rank aggregation, with implications for hiring, admissions, and information retrieval systems.

Abstract

Aggregating multiple input rankings into a consensus ranking is essential in various fields such as social choice theory, hiring, college admissions, web search, and databases. A major challenge is that the optimal consensus ranking might be biased against individual candidates or groups, especially those from marginalized communities. This concern has led to recent studies focusing on fairness in rank aggregation. The goal is to ensure that candidates from different groups are fairly represented in the top-$k$ positions of the aggregated ranking. We study this fair rank aggregation problem by considering the Kendall tau as the underlying metric. While we know of a polynomial-time approximation scheme (PTAS) for the classical rank aggregation problem, the corresponding fair variant only possesses a quite straightforward 3-approximation algorithm due to Wei et al., SIGMOD'22, and Chakraborty et al., NeurIPS'22, which finds closest fair ranking for each input ranking and then simply outputs the best one. In this paper, we first provide a novel algorithm that achieves $(2+ε)$-approximation (for any $ε> 0$), significantly improving over the 3-approximation bound. Next, we provide a $2.881$-approximation fair rank aggregation algorithm that works irrespective of the fairness notion, given one can find a closest fair ranking, beating the 3-approximation bound. We complement our theoretical guarantee by performing extensive experiments on various real-world datasets to establish the effectiveness of our algorithm further by comparing it with the performance of state-of-the-art algorithms.

Improved Rank Aggregation under Fairness Constraint

TL;DR

This work advances fair rank aggregation under proportional fairness by delivering a -approximation through a two-stage approach that reduces the problem to a colorful bi-partition on a weighted tournament, then applies PTAS-based ranking on each partition. It also introduces a generic -approximation algorithm that works with any fairness notion provided a closest fair ranking can be computed, enabling robust guarantees across stricter fairness notions like block fairness. Empirical evaluation on football and Movielens datasets shows substantial improvements over prior baselines and confirms the practical effectiveness of the proposed methods. Overall, the paper provides theoretically tight ideas and scalable algorithms that significantly improve fair rank aggregation, with implications for hiring, admissions, and information retrieval systems.

Abstract

Aggregating multiple input rankings into a consensus ranking is essential in various fields such as social choice theory, hiring, college admissions, web search, and databases. A major challenge is that the optimal consensus ranking might be biased against individual candidates or groups, especially those from marginalized communities. This concern has led to recent studies focusing on fairness in rank aggregation. The goal is to ensure that candidates from different groups are fairly represented in the top- positions of the aggregated ranking. We study this fair rank aggregation problem by considering the Kendall tau as the underlying metric. While we know of a polynomial-time approximation scheme (PTAS) for the classical rank aggregation problem, the corresponding fair variant only possesses a quite straightforward 3-approximation algorithm due to Wei et al., SIGMOD'22, and Chakraborty et al., NeurIPS'22, which finds closest fair ranking for each input ranking and then simply outputs the best one. In this paper, we first provide a novel algorithm that achieves -approximation (for any ), significantly improving over the 3-approximation bound. Next, we provide a -approximation fair rank aggregation algorithm that works irrespective of the fairness notion, given one can find a closest fair ranking, beating the 3-approximation bound. We complement our theoretical guarantee by performing extensive experiments on various real-world datasets to establish the effectiveness of our algorithm further by comparing it with the performance of state-of-the-art algorithms.
Paper Structure (33 sections, 16 theorems, 74 equations, 18 figures, 1 table, 3 algorithms)

This paper contains 33 sections, 16 theorems, 74 equations, 18 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Other related works
  4. Preliminaries
  5. Notations.
  6. Distance metric and fair rank aggregation.
  7. Weighted tournament.
  8. Colorful Bi-partition on Tournaments
  9. Description of the algorithm.
  10. Running time of Algorithm \ref{['alg:min_bi_partition']}.
  11. Approximation guarantee.
  12. Approximating the Fair Aggregate Ranking
  13. Description of the algorithm.
  14. Running time of Algorithm \ref{['alg:fair-aggregation']}.
  15. Approximation guarantee.
  16. ...and 18 more sections

Key Result

Theorem 5

There is an algorithm that, given a weighted colored tournament $T=(V,A)$ with $w:A \to \mathbb{R}$, $\texttt{col}: V \to [g]$ (for some integer $g \ge 1$), satisfying both the probability and the triangle inequality constraints, and an integer $k$, $\bar{\alpha} \in [0,1]^g$, $\bar{\beta} \in [0,1]

Figures (18)

  • Figure 1: Vertices $\{c_1, c_2, \ldots, c_d\}$ are sorted by their weighted in-degrees. For the bi-partition $(L^*, R^*)$, one of the edges in $A(y, L^*)$ is shown in cyan and one of the edges in $A(R^*, x)$ is shown in blue. For the bi-partition $(\hat{L}, \hat{R})$, the edge $(x, y)$ is shown in red. Also, one of the edges in $A(x, L^*)$ is shown in orange and one of the edges in $A(R^*, y)$ is shown in violet.
  • Figure 2: Football dataset. The $x$-axis indicates the value of the parameter ($n$, $d$, or $k$). The $y$-axis indicates the objective value of each output ranking on the left figure, with the corresponding approximation ratio on the right figures.
  • Figure 3: Movielens dataset. The x-axis indicates the value of the parameter ($n$, $d$ or $k$). The y-axis indicates the objective value of the output ranking for each algorithm.
  • Figure 4: Reduced Movielens dataset. The x-axis indicates the value of the parameter $k$. The y-axis indicates the objective value of each output ranking on the left figure, with the corresponding approximation ratio on the right figure.
  • Figure 5: Football dataset. The input instance is the title of the set of plots. The x-axis indicates the value of the parameter ($n$, $d$ or $k$). The y-axis indicates the objective value of each output ranking on the left figure, with the corresponding approximation ratio on the right figure.
  • ...and 13 more figures

Theorems & Definitions (36)

  • Definition 1: Fair Ranking
  • Definition 2: Kendall tau distance
  • Definition 3
  • Definition 4: Colorful Bi-partition Problem
  • Theorem 5
  • Lemma 6
  • proof : Proof of Lemma \ref{['lemma:optimal-bipartition-swap']}
  • proof : Proof of Theorem \ref{['thm:bi-partition']}
  • Theorem 7
  • Theorem 8
  • ...and 26 more